Physical Sciences and Mathematics Commons^™

On The Convergence Of Successive Approximations For Quasi-Nonexpansive Mappings Through Abstract Cones, B. B. Williams, J. C. Bolen

Mathematics Technical Papers

In a recent paper, Petryshyn and Williamson [4] investigated the convergence of successive approximations of quasi- nonexpansive mappings in a Banach space. This paper contains an outline, in chronological order, of the main results concerning the convergence of iteration method and consequently includes a number of references. Perov and Kibenko [3] employed generalized Banach spaces to extend contraction mapping principal and to show the flexibility of such an approach in applications. See also Bernfeld and Lakshmikantham [1]. More recently, Eisenfeld and Lakshmikantham [5,6] proved some fixed point theorems in abstract cones which extend and generalize many known results. In this …

Maximal And Minimal Solutions And Comparison Principle For Differential Equations In Abstract Cones, Roger W. Mitchell, A. Richard Mitchell, V. Lakshmikantham

Mathematics Technical Papers

Existence of maximal and minimal solutions for differential equations in abstract cones is established without requiring uniform continuity. Utilizing such a result an abstract comparison principle is developed. The results of the paper significantly improve earlier results of the authors and also simplify the proofs of other known results.

Remarks On Nonlinear Contraction And Comparison Principle In Abstract Cones, V. Lakshmikantham, Jerome Eisenfeld

Mathematics Technical Papers

The contraction mapping principle and the Schauder principle can both be viewed as a comparison of maps. For the former one has a condition of the type [see pdf for notation] and for the latter one has a condition of the type [see pdf for notation] where p is the metric and y is the Kuratowski measure of noncompactness. If p is a linear map [see pdf for notation] from the nonnegative reals [see pdf for notation] into itself then the map T satisfying (1.1) is said to be k-contractive and the map satisfying (1.2) is said to be k-set …

On The Existence Of Solutions Of Differential Equations Of Retarded Type In A Banach Space, V. Lakshmikantham, Roger W. Mitchell, A. Richard Mitchell

Mathematics Technical Papers

The existence and uniqueness of solutions of differential equations of retarded type in a Banach space are considered under a monotonicity type condition. The tools that are available for the study of equations without delay pose problems in this case since the domain and the range come from different Banach spaces. To overcome this difficulty, subsets of the domain have to be chosen carefully and weaker forms of differential inequalities have to be employed.

On A Measure Of Nonconvexity And Applications, V. Lakshmikantham, Jerome Eisenfeld

Mathematics Technical Papers

The measure of noncompactness which was introduced by Kuratowski [8] (in 1930) has now become an important tool in nonlinear analysis (although its value in that regard was not appreciated until much later). Following Kuratowski we introduce a measure of nonconvexity which has many properties in common with the measure of noncompactness and therefore we may now have "convex" where previously we had "compact" in the statements of some theorems.

Estimation Of Growth Curves By Least Square Splines, Dorothy Rybaczyk Pathak

Mathematics & Statistics ETDs

The primary object of this dissertation is to present some con­tributions to the theory of estimation of growth curves by least square splines in the presence of unknown unequal variances. The theoretical developments rest heavily on the standard least square theory and the theory of polynomial spline functions. A modifica­tion of the Aitken procedure of weighted least squares is used to estimate regression parameters. It is shown that this modification of the Aitken procedure does not unduly influence the nice least square properties of estimators so obtained; the estimators re­ main unbiased, consistent and asymptotically efficient.

The techniques developed in …

The Differentiability Of AX, J. A. Eidswick

Department of Mathematics: Faculty Publications

A "from scratch" proof of the differentiability of a^x, a > 0, is avoided by essentially all modern-day authors. A slick and popular way of handling the problem is to define a^x as e^x log a its differentiability and other properties following from that of the functions e^x and log x. Unfortunately, the usual definitions of e^x and log x involve relatively sophisticated ideas (e.g., integration or power series). Furthermore, the student, having heard of e, the natural logarithm base, at an early stage of his development, is hardly enlightened when he is …

An Evaluation Of Truncated Sequential Test, Ryh-Thinn Chang

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The development of sequential analysis has led to the proposal of tests that are more economical in that the Average Sample Number (A. S. N.) of the sequential test is smaller than the sample size of the fixed sample test. Although these tests usually have a smaller A. S. N. than the equivelent fixed sample procedure, there still remains the possibility that an extremely large sample size will be necessary to make a decision. To remedy this, truncated sequential tests have been developed.

A method of truncation for testing a composite hypotheses is studied. This method is formed by mixing …

A Simple Characterization Of Commutative Rings Without Maximal Ideals, Melvin Henriksen

All HMC Faculty Publications and Research

In a course in abstract algebra in which the instructor presents a proof that each ideal in a ring with identity is contained in a maximal ideal, it is customary to give an example of a ring without maximal ideals.

Generalizing Binary Operations, Dennis C. Smolarski

Mathematics and Computer Science

Most day to day calculations take place within the field of real numbers with the two binary operations of addition and multiplication. In this field, these two operations are definitionally independent of one another. However, if we approach binary operations from a different point of view, e.g. that of recursive formulae, we can develop multipli cation from addition by use of the concept of repeated addition. Along similar lines, we can develop exponentiation from multiplication by re peated multiplication. The next logical step would be to try to develop another binary operation based on repeated exponentiation.

Statistical Analysis Of Fourier Coefficients In A Restricted Harmonic Dial, Danny D. Dyer

Mathematics Technical Papers

Due to the physical nature of certain periodic data, the harmonic dial points (the Fourier coefficients obtained from harmonic analysis of the data) are sometimes restricted to circular regions in the dialplane. It is proposed that a circular normal distribution (CND) truncated outside a circular region be used to describe the probabilistic behavior of the random phenomena. Recurrence relations for the population moments of a CND truncated outside a circular region are derived. These recurrence relations are used to obtain consistent asymptotically (jointly) normal estimators of the unknown parameters of the distribution. A numerical example based on the harmonic dial …

Steenrod Homology And Operator Algebras, Jerome Kaminker, Claude Schochet

Mathematics Faculty Research Publications

The recent work of Larry Brown, R. G. Douglas, and Peter Fillmore on operator algebras has created a new bridge between functional analysis and algebraic topology. This note constitutes an effort to make that bridge more concrete.

What Is Number?, Willis J. Alberda

Pro Rege

No abstract provided.

Perfect Polynomials Over Gf(Q), James R. O'Connell Jr., Jacob T. B. Beard Jr., Karen I. West

Mathematics Technical Papers

A monic polynomial [see pdf for notation] is called perfect over GF(q) if and only if the sum [see pdf for notation] of the distinct monic divisors in GF[q,x] of A(x) equals A(x). Principal results characterize all perfect polynomials over GF(p) which split in GF[p,x]. Related results lead to conjectured analogs of the classical problem on the existence of odd perfect numbers.

Vector Lyapunov Functions And Perturbations Of Nonlinear Systems, Marion E. Moore, Roger W. Mitchell

Mathematics Technical Papers

Two recent papers [2,3] have combined the techniques of the Lyapunov method and the nonlinear variation of parameters to study the effects of perturbations of nonlinear differential systems. In this paper, we generalize the results obtained in [2,3] by the use of several Lyapunov functions and the concept of a generalized norm. To convince ourselves that we do have more flexibility in working with the generalized norm and vector Lyapunov functions see [1,4].

Block Diagonalization And Eigenvalues, Jerome Eisenfeld

Mathematics Technical Papers

Let A denote an Algebra with an identity element. Consider an [see pdf for notation] matrix [see pdf for notation] with a partitioning [see pdf for notation] where E and H have respective orders [see pdf for notation] and [see pdf for notation]. We seek to obtain conditions under which A is similar to a matrix D of the form [see pdf for notation] where [see pdf for notation] denotes the zero [see pdf for notation] matrix over A. Some advantage is gained in working in a general algebra. The algebra A may be taken as a Banach algebra of …

Unitary Perfect Polynomials Over Gf(Q), Jacob T. B. Beard

Mathematics Technical Papers

For monic polynomials A(x), B(x) e GF[q,x], call B(x) a unitary divisor of A(x) provided (B(x),A(x)/B(x)) = 1 . The polynomial A(x) is called unitary perfect over GF(q) if and only if the sum [see pdf for notation] of the distinct unitary divisors of A(x) equals A(x). Principal results characterize all unitary perfect polynomials over GF(p) which split in GF[p,x].

Some Combinatorial Arrangements And Incomplete Block Designs Through Them., Anis Chandra Mukherjee Dr.

Doctoral Theses

The use and importance, in Statistical Experimental, of Incomplete Block Designs, particularly, Balanced Incomplete Block (BIB) Designs, Doubly Balanced Incomplete Block (DBIB) Designs and Partially Balanced Inçomplete Block (PBIB) Designs are well known. Several combinational arrangements, including the incidence matrices of these Incomplete Block Designs and association matrices associated with PBIB Designs are known to be of use in Design of Experiments. In this thesis, we consider the construction problems pertaining to some of these combinational arrangements and take up the problem of construction of BIB, DBIB and FBIB Designs through them. The combinational arrangements studied in the thesis have, …

Generalized Inverses Of Special Types Of Matrices., Rao Vara Prasada Pullepu Sree Satya Narayana Dr.

Doctoral Theses

Inverses, in the regular sense of the term, do not exist for singular square matrices and rectangular matrices. however for such matrices there exist matrices which satisfy many important properties similar to those of inverses of nonsingular matrices and for many purposes, can be used in the same way as regular inverses. These matrices are named generalised inverses (g-inverses) to distinguish then from the inverses of nonsingular matricos. Only since 1955 this field of study af generalized inverse was: invostigated systenatically and was expiorcd for nany beautiful and interesting results and applications though the concept of generalizod inverse was first …

On A General Class Of Unbiased Ratio, Product, Ratio-Cum-Product And Regression Type Estimators., N. S. Sastry Dr.

Doctoral Theses

Probability enmpling methods ha ve extenaive a pplication in la rge- eca le agricultural and socio economic surve ys, Sampling is ale used in one or more stages of a census and sometimes as a mubstitute for a complete Census. Over the pa st four deca de s, confidence in the use of san pling and a ppreciation for the technique have stee dily grown thanks to the pi onee ring efforts of Neyman, Hansen, Hurwit z, Maha lanobis, Sukha tme, Cochran, Yate s and others.2. It is well known that the principa i problem of the theory of sampling …

Some Aspects Of Topological Semigroup Acts And Machines., Kripasindhu Sikdar Dr.

Doctoral Theses

No abstract provided.

On Some Problems Of Sequenting And Grouping., T. S. Arthanari Dr.

Doctoral Theses

The problems of sequencing arise in almost all walks of life. Theory of scheduling deals with such problems. Usually, these probl ems are stated in the literature in terms of jobs, machines, operations, penalties, due dates et cetera, that is, in the language of machine shops. The real life problems of In machine - shop job sequencing are of a complex nature. general, we consider processing n items on a certain group of machines, so as to optimize certain objective, subject to various cons traints on precedence, machine availability, due date and so on. The job sequencing problems are includ …

On The Existence Of Zeros Of Lyapunov-Monotone Operators, S. Leela, V. Lakshmikantham

Mathematics Technical Papers

Consider a nonlinear operator T from a Banach space into itself. The study of the existence of zeros of T plays an important role in yielding fixed points of nonlinear operators. The operator T has a zero if and only if the initial value problem [see pdf for notation],has a constant solution. If T is a monotone operator then (1.1) has a unique solution [see pdf for notation] defined on [see pdf for notation] and the solution operator [see pdf for notation] is nonexpansive for all [see pdf for notation]. Imposing further assumptions one can show that U(t) must have …

Univalence Of Derivatives Of Functions Defined By Gap Power Series, S. M. Shah, S. Y. Trimble

Mathematics and Statistics Faculty Research & Creative Works

No abstract provided.

A Coefficient Inequality For Convex Univalent Functions, S. Y. Trimble

Mathematics and Statistics Faculty Research & Creative Works

A short proof of formula presented is given for normalized convex univalent functions. © 1975, American Mathematical Society.

On Functional Equations Related To Mielnik's Probability Spaces, C. F. Blakemore, Caslav V. Stanojevic

Mathematics and Statistics Faculty Research & Creative Works

It is shown that the method used by C. V. Stanojevic to obtain a characterization of inner product spaces in terms of a Mielnik probability space of dimension 2 does not admit a generalization to dimension n > 2. © 1975 American Mathematical Society.

Untitled (Subject: Measuring Sound), Richard C. Heyser

Unpublished Writings

In this essay, Richard C. Heyser explains his opinions and experiences on the measurement of sound. Heyser addresses the need to standardize language when addressing sound observations and discusses subjective perceptions of sound.

The Absolute Continuity Of Phase Operators, Joanne Dombrowski, G. H. Fricke

Mathematics and Statistics Faculty Publications

This paper studies the spectral properties of a class of operators known as phase operators which originated in the study of harmonic oscillator phase. Ifantis conjectured that such operators had no point spectrum. It was later shown that certain phase operators were, in fact, absolutely continuous and that all phase operators at least had an absolutely continuous part. The present work completes the discussion by showing that all phase operators are absolutely continuous.

Multivalue Methods, Clayton V. Henrie

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Methods of solving ordinary differential equations with initial conditions are of a great importance to engineers and scientists. Many of these equations can be solved by well-known analytical techniques, but a greater number of physically significant differential equations cannot be so solved. Thus, the solutions of these equations must be approximated numerically. It is the purpose of this paper to investigate the techniques used in solving differential equations with initial conditions by "multivalue methods."

An Evaluation Of Bartlett's Chi-Square Approximation For The Determinant Of A Matrix Of Sample Zero-Order Correlation Coefficients, Stephen M. Hattori

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The single equation least-squares regression model has been extensively studied by economists and statisticians alike in order to determine the problems which arise when particular assumptions are violated. Much literature is available in terms of the properties and limitations of the model. However, on the multicollinearity problem, there has been little research, and consequently, limited literature is available when the problem is encountered. Farrar & Glauber (1967) present a collection of techniques to use in order to detect or diagnose the occurrence of multicollinearity within a regression analysis. They attempt to define multicollinearity in terms of departures from a hypothesized …